On the reducibility of moduli spaces of stable 2-bundles over an elliptic surface (Q1189281)

We consider the problem that the moduli spaces of stable 2-bundles \(E\) over a generic regular elliptic surface \(S\) with \(\hbox{det} (E)\simeq{\mathcal O}_ S\) are reducible for \(c_ 2(E)\) sufficiently large if \(S\) contains multiple fibres. Using the work of \textit{R. Friedman} [Invent. Math. 96, No. 2, 283-332 (1989; Zbl 0671.14006)] we solve this problem in the cases when \(S\) has exactly one multiple fibre.